On some properties of elliptical distributions

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On some properties of elliptical distributions

Fredrik Armerin

∗

Abstract

We look at a characterization of elliptical distributions in the case when finiteness of moments of the random vector is not assumed. Some addi-tional results regarding elliptical distributions are also presented.

Keywords: Elliptical distributions, multivariate distributions.

JEL Classification: C10.

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1 Introduction

Let the n-dimensional random vector X have finite second moments and the property that the distribution of every random variable on the form hT_{X + a for}

every h ∈ Rn

and a ∈ R is determined by its mean and variance. Chamberlain [2] showed that if the covariance matrix of X is positive definite, then this is equivalent to the fact that X is elliptically distributed. There are, however, elliptical distributions that do not have even finite first moments. In this note we show that for a random vector to be elliptically distributed is equivalent to it fulfilling a condition generalizing the moment condition above, and one that can be defined even if the random vector do not have finite moments of any order.

In portfolio analysis, if r is an n-dimensional random vector of returns and there is a risk-free rate rf, then the expected utility of having the portfolio

(w, w0) ∈ Rn× R is given by

Eu wTr + w0rf ,

where u is a utility function (we assume that this expected value is well defined). See e.g. Back [1], Cochrane [3] or Munk [5] for the underlying theory. If the distribution of wT_{r + r}

f only depends on its mean and variance, then

Eu wT_{r + w}

0rf = U wTµ + w0rf, wTΣw

(1) for some function U (this is one of the applications considered in Chamberlain [2]). If we only consider bounded utility functions, then the expected value is well defined even if r does not have any finite moments. Below we will show that if Equation (1) holds for any bounded and measurable function u, then this is in fact a defining property of elliptical distributions, i.e. r must be elliptically distributed if Equation (1) holds for any bounded and measurable u.

The results presented in this note are previously known, or are basic gener-alizations of known results.

2 Basic definitions

The general reference for this section is McNeil et al [4].

Definition 2.1 An n-dimensional random vector X has a spherical distribution if

U X = Xd

for every orthogonal n × n matrix U , i.e. for every n × n matrix U such that U UT = UTU = I.

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Theorem 2.2 Let X be an n-dimensional random vector. The following are equivalent.

(i) The random vector X has a spherical distribution. (ii) There exists a function ψ of a scalar variable such that

EheihTXi= ψ(hTh)

for every h ∈ Rn_.

(iii) For every h ∈ Rn

hTX = khkXd 1,

where khk2_{= h}T_h.

We call ψ in the theorem above the characteristic generator of X, and write X ∼ Sn(ψ) if X is n-dimensional and has a spherical distribution with characteristic

generator ψ.

For a strictly positive integer n we let (ek) for k = 1, . . . , n denote the vectors

of the standard basis in Rn_{. If X ∼ S}

n(ψ) for some characteristic generator ψ,

then by choosing first h = ek and then h = −ek we get from property (iii) in

Theorem 2.2 that

−Xk d

= Xk,

i.e. each component in a random vector which has a spherical distribution is a symmetric random variable. By choosing first h = ek and then h = e` and

again using (iii) in Theorem 2.2, we get

Xk d

= X`

for every k, ` = 1, . . . , n, i.e. every component of a spherically distributed ran-dom vector has the same distribution.

Definition 2.3 An n-dimensional random vector X is said to be elliptically distributed if

X= µ + AY,d where µ ∈ Rn_{, A is an n × k-matrix and Y ∼ S}

k(ψ). With Σ = AAT we write

X ∼ En(µ, Σ, ψ) in this case.

The characteristic function of X ∼ En(µ, Σ, ψ) is given by

EheihTXi= eihTµψ(hTΣh).

If X has finite mean, then

µ = E [X] , and if X has finite variance, then we can choose

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If X ∼ En(µ, Σ, ψ), B is an k × n-matrix and b is an k × 1-dimensional vector,

then

BX + b ∼ Ek(Bµ + b, BΣBT, ψ).

Alternatively, if X ∼ En(µ, Σ, ψ), then

BX + b= Bµ + BAY,d

where Y ∼ Sn(ψ) and AAT = Σ. Finally, when Σ is a positive definite matrix

we have the equivalence

X ∼ En(µ, Σ, ψ) ⇔ Σ−1/2(X − µ) ∼ Sn(ψ).

3 Characterizing elliptical distributions

The following proposition shows the structure of any elliptically distributed random vector.

Proposition 3.1 Let µ ∈ Rn _{and let Σ be an n × n symmetric and positive}

semidefinite matrix. For an n-dimensional random vector X the following are equivalent. (i) X ∼ En(µ, Σ, ψ). (ii) We have hTX = hd Tµ + √ hT_{Σh Z}

for any h ∈ Rn_{, where Z is a symmetric random variable with}

EeitZ = ψ(t2).

Proof. (i) ⇒ (ii): If X ∼ En(µ, Σ, ψ), then for every h ∈ Rn and some matrix

A such that AAT = Σ hTX =d hTµ + hTAY = hTµ + (ATh)TY d = hTµ + kAThkY1 = hTµ + √ hT_AAT_hY 1 = hTµ + √ hT_ΣhY 1.

Since Y has a spherical distribution, Y1 is a symmetric random variable with

characteristic function

EeitY1 = ψ(t2_).

(ii) ⇒ (i): If X has the property that

hTX = hd Tµ + √

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for every h ∈ Rn and where EeitZ = ψ(t2), then EheihTXi= eihTµEhei √ hT_{Σh Z}i = eihTµψ(hTΣh), i.e. X ∼ En(µ, Σ, ψ). 2

Note that the previous proposition is true even if Σ is only a positive semidefinite matrix.

Remark 3.2 With the same notation as in Proposition 3.1, if the random vec-tor X has the property that for every h ∈ Rn

hTX = hd Tµ + √

hT_{Σh Z}

holds and Σ has at least one non-zero diagonal element (the only case when this does not hold is when Σ = 0), then Z must be symmetric. To see this we assume, without loss of generality, that Σ11> 0. Now first choose h = e1, and

then h = −e1. We get

X1 d = µ1+ p Σ11Z and − X1 d = −µ1+ p Σ11Z respectively, or X1− µ1 √ Σ11 d = Z and −X√1− µ1 Σ11 d = Z

respectively. It follows that Z= −Z.d Using the representation

hTX = hd Tµ + √

hT_{Σh Z}

we see that the finiteness of moments of the vector X is equivalent to the finiteness of the moments of the random variable Z. This representation is also a practical way of both defining new and understanding well known elliptical distributions. When Z ∼ N (0, 1) we get the multivariate normal distribution, and when Z ∼ t(ν) we get the multivariate t-distribution with ν > 0 degrees of freedom. The multivariate t-distribution with ν ∈ (0, 1] is an example of an elliptical distribution which does not have finite mean.

Now assume that the random vector X has the property that the distribution of hT

X + a is determined by its mean and variance for every h ∈ Rn

and a ∈ R. If we let µ = E [X] and Σ = Var(X), which we assume is a positive definite matrix, then Chamberlain [2] showed that X must be elliptically distributed. Hence in this case, if

EhT

1X + a1 = E hT2X + a2

and Var(hT₁X + a1) = Var(hT2X + a2),

then we must have

hT₁X + a1 d

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This property can, with notation as above, be rewritten as follows: If hT₁µ + a1= hT2µ + a2 and hT1Σh1= hT2Σh2, then hT₁X + a1 d = hT₂X + a2.

It turns out that this condition, which is well defined for any X ∼ En(µ, Σ, ψ)

even if no moments exists, is a defining property of elliptical distributions if Σ is a positive definite matrix.

definite matrix. For an n-dimensional random vector X the following are equiv-alent.

(i) X ∼ En(µ, Σ, ψ) for some characteristic generator ψ.

(ii) For any measurable and bounded f : R → R and any h ∈ Rn

and a ∈ R Ef (hTX + a) = F (hTµ + a, hTΣh)

for some function F : R × R+→ R.

(iii) If

hT₁µ + a1= hT2µ + a2 and hT1Σh1= hT2Σh2

for h1, h2∈ Rn and a1, a2∈ R, then

hT₁X + a1 d

= hT₂X + a2.

For a proof of this, see Section A.1. It is possible to reformulate this proposition without using the constants a, a1 and a2.

definite matrix. For an n-dimensional random vector X the following are equiv-alent.

(i) X ∼ En(µ, Σ, ψ) for some characteristic generator ψ.

(ii) For any measurable and bounded g : R → R and any h ∈ Rn

Eg(hT(X − µ)) = G(hTΣh) for some function G : R+→ R.

(iii) If

hT₁Σh1= hT2Σh2

for h1, h2∈ Rn, then

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For a proof, see Section A.2.

In Propositions 3.3 and 3.4 we assumed that the matrix Σ was positive def-inite. The implications (i) ⇒ (ii) and (ii) ⇒ (iii) in these propositions are still valid when Σ is only positive semidefinite (and the general characterization of elliptical distributions in Proposition 3.1 also holds in this case). The implica-tions (iii) ⇒ (i) in the proposiimplica-tions above are not true in general, as is seen in the following example.

Example 3.5 Let Σ = 1 0 0 0 and X = U 0 ,

where U ∼ N (0, 1). In this case we have

hT₁Σh1= hT2Σh2 ⇒ hT1X d

= hT₂X,

so X has property (iii) from Proposition 3.4. But X is not spherically dis-tributed. This follows from the fact that every component of a spherically distributed random vector must have the same distribution.

By letting µ = [0 0]T it is possible to also construct a counterexample to the implication (iii) ⇒ (i) in Proposition 3.3. 2

A

Proofs

A.1 Proof of Proposition 3.3

(i) ⇒ (ii): We know that there exists a symmetric random variable Z such that

hTX = hd Tµ + √

hT_{Σh Z}

for any h ∈ Rn. Hence for any measurable and bounded f

Ef (hT_{X + a) = E}h_f_hT_{µ +}√_hT_{Σh Z + a}i_{= F h}T_{µ + a, h}T_{Σh ,}

where

F (x, y) = E [f (x +√y Z)] .

(ii) ⇒ (iii): Fix t ∈ R and let f1(x) = sin tx and f2(x) = cos tx (which are two

bounded and measurable functions). Define Fi, i = 1, 2, by

Efi(hTX + a) = Fi(hTµ + a, hTΣh).

Now take any h1, h2∈ Rn and a1, a2∈ R such that

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Then

Eheit(hT1X+a1)i ₌ _Esin(t(hT

1X + a1)) + i cos(t(hT1X + a1)) = F1(hT1µ + a1, hT1Σh1) + iF2(hT1µ + a1, hT1Σh1) = F1(hT2µ + a2, hT2Σh2) + iF2(hT2µ + a2, hT2Σh2) = Esin(t(hT 2X + a2)) + i cos(t(hT2X + a2)) = Eheit(hT2X+a2)i_.

Since this holds for any t ∈ R we have

hT₁X + a1 d

= hT₂X + a2.

(iii) ⇒ (i): Take h ∈ Rn _{and let}

h1 = Σ−1/2h a1 = −hTΣ−1/2µ and h2 = khkΣ−1/2e1 a2 = −khkeT1Σ−1/2µ. Then hT₁Σh1= khk2 and hT2Σh2= khk2. We also have hT₁µ + a1= hTΣ−1/2µ + (−hTΣ−1/2µ) = 0 and hT₂µ + a2= khkeT1Σ −1/2_{µ + (−khke}T 1Σ −1/2_{µ) = 0.} It follows that hT1X + a1 d = hT2X + a2 ⇔ hTΣ−1/2(X − µ)= khked T₁Σ−1/2(X − µ). This shows that

Σ−1/2(X − µ) ∼ Sn(ψ),

which, since Σ is a positive definite matrix, is equivalent to X ∼ E(µ, Σ, ψ).

A.2 Proof of Proposition 3.4

(i) ⇒ (ii): There exists a symmetric random variable Z such that

hTX = hd Tµ + √

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for any h ∈ Rn. It follows that for any bounded and measurable g

Eg(hT_{(X − µ)) = E}h_g√_hT_{Σh Z}i_{= G h}T_{Σh ,}

where

G(x) = Eg(√x Z) .

(ii) ⇒ (iii): Fix t ∈ R and let g1(x) = sin tx and g2(x) = cos tx (which are two

bounded and measurable functions). Define Gi, i = 1, 2, by

Egi(hT(X − µ)) = Gi(hTΣh).

Now take any h1, h2 such that

hT₁Σh1= hT2Σh2. Then EheithT1(X−µ) i = Esin(thT 1(X − µ)) + i cos(th T 1(X − µ)) = G1(hT1Σh1) + iG2(hT1Σh1) = G1(hT2Σh2) + iG2(hT2Σh2) = Esin(thT 2(X − µ)) + i cos(th T 2(X − µ)) = EheithT2(X−µ) i .

Since this holds for any t ∈ R we have hT

1(X − µ) d

= hT

2(X − µ).

(iii) ⇒ (i): Take h ∈ Rn _{and let}

h1= Σ−1/2h and h2= khkΣ−1/2e1, . Then hT₁Σh1= khk2 and hT₂Σh2= khk2. Hence hT₁Σh1= hT2Σh2

and it follows that

hT₁(X − µ)= hd T₂(X − µ) ⇔

hTΣ−1/2(X − µ)= khked T₁Σ−1/2(X − µ).

Since Σ is a positive definite matrix, this shows, as in the proof of Proposition 3.3, that

X ∼ En(µ, Σ, ψ).

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References

[1] Back, K. & E. (2010), “Asset Pricing and Portfolio Choice Theory”, Oxford University Press.

[2] Chamberlain, C. (1983), “A Characterization of the Distributions That Im-ply Mean-Variance Utility functions”, Journal of Economic Theory 29, p. 185-201.

[3] Cochrane, J. H. (2001), “Asset Pricing”, Princeton University Press. [4] McNeil, A. J., Frey, R. & Embrechts P. (2005), “Quantitative Risk

Man-agement”, Princeton University Press.

[5] Munk, C. (2013), “Financial Asset Pricing Theory”, Oxford University Press.